Question

Solve the separable equation using linear substitution: y^,=4/(x+y)²

Answer 1

Let us solve the separable equation $y'=\frac{4}{(x+y)^2}$ using the linear substitution $z=x+y.$ It follows that $z'=1+y',$ and hence we get the equation $z'-1=\frac{4}{z^2}.$ The last equation is equivalent to $\frac{dz}{dx}=\frac{4}{z^2}+1,$ and hence to $dx=\frac{z^2dz}{z^2+4}.$ It follows that

$\int dx=\int\frac{z^2dz}{z^2+4} =\int\frac{(z^2+4-4)dz}{z^2+4} =\int dz-4\int\frac{dz}{z^2+4}=z-2\arctan\frac{z}2+C.$

Therefore, $x=z-2\arctan\frac{z}2+C.$

We conclude that the general solution of the differential equation $y'=\frac{4}{(x+y)^2}$ is

$x=x+y-2\arctan\frac{x+y}2+C$ or $y=2\arctan\frac{x+y}2+C.$